Optimal. Leaf size=585 \[ -\frac{i \sqrt{f} p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 g^{3/2}}-\frac{i \sqrt{f} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 g^{3/2}}+\frac{i \sqrt{f} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{3/2}}-\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}+\frac{x \log \left (c \left (d+e x^2\right )^p\right )}{g}+\frac{\sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{g^{3/2}}+\frac{\sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{g^{3/2}}+\frac{2 \sqrt{d} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g}-\frac{2 \sqrt{f} p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{g^{3/2}}-\frac{2 p x}{g} \]
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Rubi [A] time = 0.594126, antiderivative size = 585, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {2476, 2448, 321, 205, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ -\frac{i \sqrt{f} p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 g^{3/2}}-\frac{i \sqrt{f} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 g^{3/2}}+\frac{i \sqrt{f} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{3/2}}-\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}+\frac{x \log \left (c \left (d+e x^2\right )^p\right )}{g}+\frac{\sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{g^{3/2}}+\frac{\sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{g^{3/2}}+\frac{2 \sqrt{d} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g}-\frac{2 \sqrt{f} p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{g^{3/2}}-\frac{2 p x}{g} \]
Antiderivative was successfully verified.
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Rule 2476
Rule 2448
Rule 321
Rule 205
Rule 2470
Rule 12
Rule 4928
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx &=\int \left (\frac{\log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{g \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{\int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{g}-\frac{f \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx}{g}\\ &=\frac{x \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}-\frac{(2 e p) \int \frac{x^2}{d+e x^2} \, dx}{g}+\frac{(2 e f p) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g} \left (d+e x^2\right )} \, dx}{g}\\ &=-\frac{2 p x}{g}+\frac{x \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}+\frac{\left (2 e \sqrt{f} p\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{d+e x^2} \, dx}{g^{3/2}}+\frac{(2 d p) \int \frac{1}{d+e x^2} \, dx}{g}\\ &=-\frac{2 p x}{g}+\frac{2 \sqrt{d} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g}+\frac{x \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}+\frac{\left (2 e \sqrt{f} p\right ) \int \left (-\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{g^{3/2}}\\ &=-\frac{2 p x}{g}+\frac{2 \sqrt{d} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g}+\frac{x \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}-\frac{\left (\sqrt{e} \sqrt{f} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{-d}-\sqrt{e} x} \, dx}{g^{3/2}}+\frac{\left (\sqrt{e} \sqrt{f} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{-d}+\sqrt{e} x} \, dx}{g^{3/2}}\\ &=-\frac{2 p x}{g}+\frac{2 \sqrt{d} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g}-\frac{2 \sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{3/2}}+\frac{\sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{3/2}}+\frac{\sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{3/2}}+\frac{x \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}+2 \frac{p \int \frac{\log \left (\frac{2}{1-\frac{i \sqrt{g} x}{\sqrt{f}}}\right )}{1+\frac{g x^2}{f}} \, dx}{g}-\frac{p \int \frac{\log \left (\frac{2 \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{f} \left (-i \sqrt{e}+\frac{\sqrt{-d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{g}-\frac{p \int \frac{\log \left (\frac{2 \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{f} \left (i \sqrt{e}+\frac{\sqrt{-d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{g}\\ &=-\frac{2 p x}{g}+\frac{2 \sqrt{d} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g}-\frac{2 \sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{3/2}}+\frac{\sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{3/2}}+\frac{\sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{3/2}}+\frac{x \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}-\frac{i \sqrt{f} p \text{Li}_2\left (1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 g^{3/2}}-\frac{i \sqrt{f} p \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 g^{3/2}}+2 \frac{\left (i \sqrt{f} p\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{i \sqrt{g} x}{\sqrt{f}}}\right )}{g^{3/2}}\\ &=-\frac{2 p x}{g}+\frac{2 \sqrt{d} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g}-\frac{2 \sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{3/2}}+\frac{\sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{3/2}}+\frac{\sqrt{f} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{3/2}}+\frac{x \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac{\sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}+\frac{i \sqrt{f} p \text{Li}_2\left (1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{3/2}}-\frac{i \sqrt{f} p \text{Li}_2\left (1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 g^{3/2}}-\frac{i \sqrt{f} p \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 g^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.292048, size = 680, normalized size = 1.16 \[ \frac{i \sqrt{f} p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-i \sqrt{-d} \sqrt{g}}\right )+i \sqrt{f} p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+i \sqrt{-d} \sqrt{g}}\right )-i \sqrt{f} p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}+i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-i \sqrt{-d} \sqrt{g}}\right )-i \sqrt{f} p \text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}+i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+i \sqrt{-d} \sqrt{g}}\right )-2 \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )+2 \sqrt{g} x \log \left (c \left (d+e x^2\right )^p\right )+i \sqrt{f} p \log \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )+i \sqrt{f} p \log \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}-i \sqrt{e} \sqrt{f}}\right )-i \sqrt{f} p \log \left (1+\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}-i \sqrt{e} \sqrt{f}}\right )-i \sqrt{f} p \log \left (1+\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )+\frac{4 \sqrt{d} \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-4 \sqrt{g} p x}{2 g^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.555, size = 746, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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